A result on the equation $x^p+y^p=z^r$ using Frey abelian varieties

Abstract

We prove a diophantine result on generalized Fermat equations of the form $x^p+y^p=z^r$ which for the first time requires the use of Frey abelian varieties of dimension $\ge 2$ in Darmon's program. For that, we provide an irreducibility criterion for the mod $\mathfrak{p}$ representations attached to certain abelian varieties of ${\text{GL}}_2$-type over totally real fields.